2It is interesting to note that the integral form of Gauss' law can be used to solve for the E or Din problems where there is a high degree of symmetry (typically 2 degrees of symmetry). The next three examples demonstrate the application of Gauss' law to find the E associated with three different charge distributions. Example Find the electric field due to a uniformly distributed sphere of charge (ρVis a constant for ra≤). Solution: Using Gauss' law VVSddV⋅=∫∫∫∫∫DSwBy symmetry ˆrD=Dr, where Dris only a function of r. Thus, assuming ra≥

322000002322222ˆsinsin4sin3sin44aVrrrVrrrrddrdrddaDrd dQDrQQDrππθθφρπθθφ ρθθφ=⋅=====∫∫∫∫∫or EarrV=ε323. Note that this is the same D field as that for a point charge Qat the origin. For ra≤, 32ˆsinsin4433rVVrVrdrdrddrrDρπ===or EQrar=43πε.

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4Example Find the electric field due to an infinite uniform line charge. Solution: Using Gauss' law Sddρ⋅=∫∫∫DSAAAwBy symmetry ˆD=Dρ, where Dis only a function of . Thus, 123ˆˆˆ()()zhSSSdsdsdsdρρ==−=+⋅+⋅−+⋅=∫∫∫∫∫∫∫DρDzDzAAA20(2)222hhDdzddzDhhEπρφπρπερ−−===∫∫∫AAAThis solution agrees with our previous result.

5Static Charges and Conductors • Within and on conductors, charges can move with little resistance to motion. • If free charges are placed within a conductor there will be an electric field established between them.

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